= Graphical Lasso column regression
{c}
{title2=$b=-\widehat\Omega_{-j,j}/\widehat\Omega_{jj}$}
For a <Graphical Lasso> solution with inverse $\widehat\Sigma$, the unique minimizer of $\tfrac12\|Wb-W^{-1}S_{-j,j}\|_2^2+\lambda\|b\|_1$, $W^2=\widehat\Sigma_{-j,-j}\succ0$, is $b=-\widehat\Omega_{-j,j}/\widehat\Omega_{jj}$. The inverse identity gives $\widehat\Sigma_{-j,j}=\widehat\Sigma_{-j,-j}b$, and the off-diagonal <Graphical-Lasso Karush-Kuhn-Tucker conditions> become the column <Lasso> conditions because the corresponding signs are opposite.
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